Let $A_{ijl}(t,x) : [0,\infty) \times \mathbb{R}^n \to [\mathbb{R}^n]^3$ be a smooth tensor field. That is, $i,j,l \in \{1,2,3, \cdots, n\}$

Further assume that $A_{ijl}(t,x)=A_{jil}(t,x)$ for all $(t,x) \in [0,\infty) \times \mathbb{R}^n$.

Also, let $0=\lambda_1< \lambda_2 < \cdots < \lambda_n$ be some fixed numbers and finally suppose that $A_{ijl}(t,x)$ satisfies the following ODE: \begin{equation} \partial_t \sum_{l=1}^n A_{ijl}(t,x)= -\sum_{l=1}^n \lambda^2_l A_{ijl}(t,x) \text{ with } \sum_{l=1}^n A_{ijl}(0,x)=\delta_{ij} \end{equation}

Then, at least when $t=0$, we have \begin{equation} \partial_t \sum_{l=1}^n A_{ssl}(t,x)\mid_{t=0}=-\lambda^2_s \end{equation} for all $s=1,2, \cdots, n$ while \begin{equation} \partial_t \sum_{l=1}^n A_{ijl}(t,x)\mid_{t=0}=0 \end{equation} for $i \neq j$.

From such information, is it possible to conclude, together with the smoothness assumption, that \begin{equation} \sum_{l=1}^n A_{ijl}(t,x)=e^{-\lambda_i^2 t} \text{ if } i=j \text{ and } 0 \text{ otherwise} \end{equation} for all $t \geq 0$?

It seems quite plausible for me, but I cannot really justify my guess for extrapolation to $t>0$. Could anyone please help me?

Edit : I add one more condition that $A_{ijl}(0,x)=\delta_{il}\delta_{jl}$ so that the above explanation makes sense.